Heating, ventilation and air conditioning (HVAC) system and method using feedback linearization

ABSTRACT

A new feedback linearization approach to advanced control of single-unit and multi-unit HVAC systems is described. In accordance with the approach of the invention, this new nonlinear control includes a model-based feedback linearization part to compensate for the nonlinearity in the system dynamics. Therefore, the evaporating temperature and superheat values can be controlled by linear PI control designs to achieve desired system performance and reliability. The main advantages of the new nonlinear control approach include (1) better performance even with large model errors, (2) being able to adapt to indoor unit turn on/off operation, (3) much smaller PI control gains compared to that of current feedback PI controls, (4) much easier design procedures since there is no need for tuning the PI control gains over wide range operation.

RELATED APPLICATION

[0001] This application is based on U.S. Provisional Patent Applicationserial No. 60/442,465, filed on Jan. 23, 2003, the contents of which areincorporated herein in their entirety.

BACKGROUND OF THE INVENTION

[0002] HVAC systems for buildings are a major consumer of electricalenergy. Controlling such systems efficiently and effectively in thepresence of dynamic interaction and random disturbance so as to conserveenergy while maintaining the desired thermal comfort level requires morethan a conventional methodology. With increasing complexity of modemHVAC systems such as multi split systems (Variable Refrigerant Volumesystems with multiple evaporators), controlling and optimizing theoperation with guaranteed performance, stability and reliability becomesa challenging issue. In addition, complex HVAC systems have a variety ofsystem configurations to meet customers' needs, different operatingconditions, and varying environmental conditions. Innovative controldesign is needed to provide desired system performance and reliabilityand to dramatically reduce the time and cost of control design processesfor complex HVAC systems.

SUMMARY OF THE INVENTION

[0003] The invention is directed to a heat transfer system and method.In accordance with the approach of the invention, the system includes afirst heat exchanger and a second heat exchanger in thermalcommunication with a space. A processor estimates an amount of heattransferred between the second heat exchanger and the space and alters acontrol parameter of the heat transfer system based on the estimatedamount of heat transferred to control the heat transfer system.

[0004] In one embodiment, the first heat exchanger is a condenser.Alternatively, the first heat exchanger is an evaporator. The secondheat exchanger can be a condenser. Alternatively, the second heatexchanger can be an evaporator.

[0005] In one embodiment, the processor controls a temperature ofrefrigerant in the evaporator the processor can control a temperature ofrefrigerant in the first heat exchanger. The processor can control adegree of superheat in the evaporator. The parameter altered by theprocessor can be an expansion valve opening.

[0006] The processor can control a discharge pressure of refrigerant ina compressor of the heat transfer system. The processor can control adischarge temperature of refrigerant in a compressor of the heattransfer system.

[0007] In one embodiment, the system includes a plurality of evaporatorsin thermal communication with the space and/or a respective plurality ofspaces.

[0008] In one embodiment, the system of the invention also includes acompressor for increasing pressure of a refrigerant flowing between thefirst and second heat exchangers. The parameter altered by the processorcan be a speed of the compressor.

[0009] In one embodiment, the processor controls a temperature ofrefrigerant in the second heat exchanger. In one embodiment, theprocessor controls a temperature of refrigerant in the first heatexchanger. In one embodiment, the processor controls a degree ofsuperheat in the second heat exchanger. The parameter altered by theprocessor can be an expansion valve opening.

[0010] In one embodiment, the processor controls a discharge pressure ofrefrigerant in a compressor of the heat transfer system.

[0011] In one embodiment, the processor controls a discharge temperatureof refrigerant in a compressor of the heat transfer system.

[0012] In one embodiment, the processor controls the parameter using afeedback linearization approach.

[0013] In one embodiment, a plurality of second heat exchangers inthermal communication with the space and/or a respective plurality ofspaces.

[0014] In one embodiment, the heat transfer system is controlled toprotect a component of the heat transfer system from damage. Theprotected component can be a compressor.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] The foregoing and other objects, features and advantages of theinvention will be apparent from the more particular description of apreferred embodiment of the invention, as illustrated in theaccompanying drawings in which like reference characters refer to thesame parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingthe principles of the invention.

[0016]FIG. 1 contains a schematic diagram of a low-order evaporatormodel in accordance with the invention.

[0017]FIG. 2 contains a schematic diagram of an evaporator model formulti-systems.

[0018]FIG. 3 contains a schematic diagram of the nonlinear controlapproach in accordance with the invention.

[0019]FIG. 4 contains a block diagram of new nonlinear control approachof the invention.

[0020]FIG. 5 contains a block diagram of a linear feedback PI controlsystem.

[0021]FIG. 6A is a graph of time response of evaporating temperaturewith same PI gains for the feedback linearization and PI control of theinvention compared to feedback PI control.

[0022]FIG. 6B is a graph illustrating time response of mass flow ratewith same PI gains for the feedback linearization and PI control of theinvention compared to feedback PI control.

[0023]FIG. 6C is a graph illustrating time response of compressor speedwith same PI gains for the feedback linearization and PI control of theinvention compared to feedback PI control.

[0024]FIG. 7A is a graph of time response of evaporating temperaturewith similar performance for the feedback linearization and PI controlof the invention compared to feedback PI control.

[0025]FIG. 7B is a graph of time response of mass flow rate with similarperformance for the feedback linearization and PI control of theinvention compared to feedback PI control.

[0026]FIG. 7C is a graph of time response of compressor speed withsimilar performance for the feedback linearization an PI control of theinvention compared to feedback PI control.

[0027]FIG. 8A is a graph of time response of evaporating temperaturewhere feedback linearization is assumed to have 20% error for same PIgains.

[0028]FIG. 8B is a graph of time response of compressor speed wherefeedback linearization is assumed to have 20% error for same PI gains.

[0029]FIG. 9A is a graph of time response of evaporating temperaturewhere feedback linearization is assumed to have 20% error for similarperformance. FIG. 9B is a graph of time response of compressor speedwhere feedback linearization is assumed to have 20% error for similarperformance.

[0030]FIG. 10A is a graph of time response of evaporating temperaturewhere feedback linearization is assumed to have 20% error for same gainsand one indoor unit is turned off at t=40s and turned on at t=80s.

[0031]FIG. 10B is a graph of time response of compressor speed wherefeedback linearization is assumed to have 20% error for same gains andone indoor unit is turned off at t=40s and turned on at t=80s.

[0032]FIG. 11A is a graph of time response of evaporating temperaturewhere feedback linearization is assumed to have no estimation error andthe second indoor unit is turned off at t=40s and turned on at t=80s.

[0033]FIG. 11B is a graph of the control output of compressor speedwhere feedback linearization is assumed to have no estimation error andthe second indoor unit is turned off at t=40s and turned on at t=80s.

[0034]FIG. 12 contains a schematic block diagram of nonlinear control ofexpansion valve for l(t) in accordance with the invention.

[0035]FIG. 13 contains a schematic block diagram of nonlinear control ofexpansion valve for SH in accordance with the invention.

[0036]FIG. 14A is a graph of the time response of the length l(t) of thetwo-phase section controlled from an initial value of 6 m to the desiredvalue of 7.32 m, assuming there is no estimation error for feedbacklinearization.

[0037]FIG. 14B is a graph of control input of mass flow rate ofexpansion valve for the case of FIG. 14A.

[0038]FIG. 14C is a graph of control output of expansion valve openingfor the case of FIG. 14A.

[0039]FIG. 14D is a graph of control output of heat flow for the case ofFIG. 14A.

[0040]FIG. 14E is a graph of control output of superheat for the case ofFIG. 14A.

[0041]FIG. 15A is a graph of the time response of the length l(t) of thetwo-phase section controlled at the desired value of 7.32 m, assumingthere is 20% estimation error for feedback linearization.

[0042]FIG. 15B is a graph of control input of mass flow rate ofexpansion valve for the case of FIG. 15A.

[0043]FIG. 15C is a graph of control output of expansion valve openingfor the case of FIG. 15A.

[0044]FIG. 15D is a graph of control output of heat flow for the case ofFIG. 15A.

[0045]FIG. 15E is a graph of control output of superheat for the case ofFIG. 15A.

[0046]FIG. 16A is a graph of time response of the length l(t) of thetwo-phase section from and initial value of 6 m to the desired value of7.32 m, assuming three different load situations and no estimation errorfor feedback linearization.

[0047]FIG. 16B is a graph of control output of expansion valve openingfor the case of FIG. 16A.

[0048]FIG. 17A is a graph of control output of superheat controlled froman initial value of 11.3 C to the desired value of 5 C , assuming thatthere is no estimation error for feedback linearization.

[0049]FIG. 17B is a graph of control output of expansion valve openingfor the case of FIG. 17A.

[0050]FIG. 18 contains a schematic diagram of the low-order evaporatormodel.

[0051]FIG. 19 contains a schematic diagram of a multi-unit system inaccordance with the present invention.

[0052]FIG. 20 is a graph of inlet mass flow rate input.

[0053]FIG. 21 is a graph of outlet mass flow rate.

[0054]FIG. 22 is a graph of evaporating temperature assuming ameasurement output value of T_(e).

[0055]FIG. 23 is a graph of wall temperate Tw showing a comparison of Twfrom the model and the nonlinear observer in accordance with theinvention.

[0056]FIG. 24 is a graph of two-phase length l showing a comparison of lfrom the model and the nonlinear observer in accordance with theinvention.

[0057]FIG. 25 is a graph of response of superheat with increase ofindoor fan speed, controlling superheat from 5 C to a desired value 3.5C for the feedback linearization and PI control of the invention andfeedback PI control.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

[0058] Multi-unit HVAC systems operate in a very nonlinear way atdifferent operating conditions and ambient conditions. Duringmulti-system operation, it is very common that some units are turned onand turned off. There also exists significant nonlinearity in systemcomponents such as compressor, expansion value, and heat exchangers. Todesign a reliable and effective controller for different types ofmulti-systems is very time-consuming task. Basically, PI gains infeedback controllers currently used in actual multi-unit machines haveto be tuned to guarantee performance and stability.

[0059] Modeling of dynamic behavior of a complicated air conditioningsystem with multiple indoor units is a challenging problem. However, itis essential for improvement of design and control of such systems.Modeling of an air condition system with a single indoor unit has beenreported in several publications.

[0060] A low-order model approach in accordance with the invention isdescribed herein for multi-evaporators in a multi-unit HAVC system. Themodel describes the dynamic relation between the evaporating temperatureand compressor side mass flow rate that can be further related tocompressor speed. The model also describes the dynamic relation betweenthe length of the two-phase section of an evaporator and expansion valveside mass flow rate that can be further related to expansion valveopening. With the low-order model, a nonlinear control design methodcalled feedback linearization can be applied to compensate for thenonlinearity in the dynamics. Although the invention is described interms of a multi-unit HVAC system, it will be understood that theinvention is applicable to single-unit systems as well.

[0061] The new feedback linearization approach described herein employsmuch easier design procudures and can achieve better control performancefor wide range operation including indoor units being turned on/off.Since the nonlinearity in the system dynamics is compensated by thefeedback linearization, a PI controller design approach for a knownlinear system can be applied. The simulations demonstrate that even withlarge estimation error, the new nonlinear control of the invention canstill achieve desired performance.

1. Low-Order Evaporator Model

[0062] This section describes a low-order evaporator model that will beused for the new nonlinear control approach of the invention.

[0063] 1.1 Derivations of Equations

[0064]FIG. 1 contains a schematic diagram of a low-order evaporatormodel in accordance with the invention. Referring to FIG. 1, T_(e) (t)is evaporating temperature, l(t) is the length of the two-phase section.{dot over (m)}_(in) and {dot over (m)}_(out) are the inlet and outletrefrigerant mass flow rates, respectively. {dot over (m)}_(mid) is therefrigerant mass flow rate at the liquid dry-out point. h_(in) is theinlet refrigerant specific enthalpy. It is assumed that the two-phasesection has invariant mean void fraction {overscore (γ)}.

[0065] The mass balance equation for the two-phase section is$\begin{matrix}{{\frac{}{t}\left\lbrack {\left( {{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)} + {\rho_{g}\overset{\_}{\gamma}}} \right){{Al}(t)}} \right\rbrack} = {{\overset{.}{m}}_{in} - {\overset{.}{m}}_{mid}}} & (1)\end{matrix}$

[0066] where A is the evaporator heat exchanger pipe inner cross sectionarea, ρ_(l) and ρ_(g) are refrigerant saturated liquid and vapordensities, respectively.

[0067] The energy balance equation for the two-phase section is$\begin{matrix}{{\frac{}{t}\left\lbrack {\left( {{\rho_{l}{h_{l}\left( {1 - \overset{\_}{\gamma}} \right)}} + {\rho_{g}h_{g}\overset{\_}{\gamma}}} \right){{Al}(t)}} \right\rbrack} = {q + {{\overset{.}{m}}_{in}h_{in}} - {{\overset{.}{m}}_{mid}h_{g}}}} & (2)\end{matrix}$

[0068] where h_(l) and h_(g) are refrigerant saturated liquid and vaporspecific enthalpies and q is the evaporator heat transfer rate.

[0069] 1.2 Equation for the Two-phase Section Length

[0070] From Equations (1) and (2), the following equation is obtained byneglecting the variation of the refrigerant properties over the timestep: $\begin{matrix}{\left. {{{Equ}(2)} - {h_{g}*{{Equ}(1)}}}\Rightarrow{{\rho_{l}\left( {h_{l} - h_{g}} \right)}\left( {1 - \overset{\_}{\gamma}} \right)A\frac{{l(t)}}{t}} \right. = {q + {{\overset{.}{m}}_{in}\left( {h_{in} - h_{g}} \right)}}} & (3)\end{matrix}$

[0071] Since h_(g)−h_(l)=h_(lg), h_(in)−h_(g)=−h_(lg)(1−x₀) where x₀ isthe inlet vapor quality, Equation (3) can be expressed by$\begin{matrix}{{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}A\frac{{l(t)}}{t}} = {{- \frac{q}{h_{\lg}}} + {{\overset{.}{m}}_{in}\left( {1 - x_{0}} \right)}}} & (4)\end{matrix}$

[0072] q/h_(lg) represents the rate of liquid evaporating into vapor,and {dot over (m)}_(in) (1−x₀) is the inlet liquid mass flow rate,therefore Equation (4) represents the liquid mass balance in thetwo-phase section of evaporator.

[0073] From Equation (4), the following equation is obtained:$\begin{matrix}{\frac{{l(t)}}{t} = {{{- \frac{1}{\tau}}{l(t)}} + {\frac{\left( {1 - x_{0}} \right)}{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}A}{\overset{.}{m}}_{in}}}} & (5)\end{matrix}$

[0074] where $\begin{matrix}{\tau = \frac{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}{Ah}_{\lg}}{{\overset{\_}{q}}_{l}}} & (6)\end{matrix}$

[0075] and {overscore (q)}_(l) is evaporator heat flux per unit oflength. τ can be thought of as representing the time required toevaporate the liquid in the two-phase region.

[0076] 1.3 Equation for the Evaporating Temperature

[0077] The vapor mass balance in an evaporator is now considered. Theinlet vapor mass flow rate is {dot over (m)}_(in)x_(o), and the outletvapor mass flow rate is {dot over (m)}_(out) out when superheat ispresented. The rate of vapor generated from liquid during theevaporating process in the two-phase section is q/h_(lg). The vapor masschange with respect to time should be equal to inlet vapor mass flowrate plus the rate of vapor generated from liquid minus the outlet vapormass flow rate. Therefore $\begin{matrix}{\frac{M_{v}}{t} = {{V\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}\frac{T_{e}}{t}} = {{{\overset{.}{m}}_{in}x_{0}} + \frac{q}{h_{\lg}} - {\overset{.}{m}}_{out}}}} & (7)\end{matrix}$

[0078] where M_(ν)and V are the total vapor mass and total volume of thelow-pressure side. Te is evaporating temperature. In Equation (7), it isassumed that in the low-pressure side the vapor volume is much largerthan the liquid volume.

[0079] 1.4 Equations of Evaporator Model for Multi-systems

[0080]FIG. 2 contains a schematic diagram of an evaporator model formulti-systems. Based on Equation (4), liquid mass balance for i-thevaporator can be expressed by $\begin{matrix}{{{\rho_{l}\left( {1 - {\overset{\_}{\gamma}}_{i}} \right)}A_{i}\frac{{l_{i}(t)}}{t}} = {{- \frac{q_{i}}{h_{\lg}}} + {{\overset{.}{m}}_{{in},i}\left( {1 - x_{0}} \right)}}} & (8)\end{matrix}$

[0081] where i=1,2, . . . , n, n is the number of indoor units of amulti-system. q_(i) is the i-th evaporator heat transfer rate.

[0082] The vapor mass balance equation for all n evaporators is$\begin{matrix}\begin{matrix}{{\sum\limits_{i = 1}^{n}\quad \frac{M_{v,i}}{t}} = {{\sum\limits_{i = 1}^{n}\quad {V_{i}\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}\frac{T_{e}}{t}}} = {{\sum\limits_{i = 1}^{n}\quad {{\overset{.}{m}}_{{in},i}x_{0}}} +}}} \\{{\frac{\sum\limits_{i = 1}^{n}\quad q_{i}}{h_{\lg}} - {\overset{.}{m}}_{out}}}\end{matrix} & (9)\end{matrix}$

[0083] where ΣV_(i) is the total volume of the low-pressure side of amulti-system.

2. Innovative Nonlinear Control of Evaporating Temperature

[0084] 2.1 Nonlinear Control of Evaporating Temperature

[0085] Equation (9) can be rewritten as $\begin{matrix}{{k\frac{T_{e}}{t}} = {{\sum\limits_{i = 1}^{n}\quad {{\overset{.}{m}}_{{in},i}x_{0}}} + \frac{\sum\limits_{i = 1}^{n}\quad q_{i}}{h_{\lg}} - {\overset{.}{m}}_{out}}} & (10) \\{k = {\sum\limits_{i = 1}^{n}\quad {V_{i}\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}}}} & (11)\end{matrix}$

[0086] Equation (10) represents the vapor mass balance in thelow-pressure side of a multi-system. In the right side of Equation (10),the first term is the inlet vapor mass flow rate for all n evaporators,the second term represents the rate of vapor generated from all nevaporators, and the third term represents the outlet vapor mass flowrate. The change of evaporating temperature with respect to time isdependent on how much vapor flowing into all evaporators, how much thevapor generated from the liquid during evaporation, and how much vaporflowing into compressor.

[0087] Assuming that the outlet vapor mass flow rate {dot over(m)}_(out) is the control input and it is desired to control theevaporating temperature to a desired value T_(e,d). The control isdesigned as follows: $\begin{matrix}{{\overset{.}{m}}_{out} = {{\sum\limits_{i = 1}^{n}\quad {{\overset{.}{m}}_{{i\quad n},i}x_{o}}} + \frac{\sum\limits_{i = 1}^{n}q_{i}}{h_{\lg}} + {\frac{k}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {{kk}_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}}} & (12)\end{matrix}$

[0088] By inserting Equation (12) into Equation (10), the controlledevaporating temperature dynamics can be described by the followingequation, $\begin{matrix}{\frac{T_{e}}{t} = {{{- \frac{1}{\tau_{d}}}\left( {T_{e} - T_{e,d}} \right)} - {k_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}}} & (13)\end{matrix}$

[0089] where τ_(d) and k_(i) are design parameters representing how fastthe evaporating temperature reaches the desired set-point withguaranteed zero steady state error. From Equation (13) we can see thatthe closed loop dynamics is linear and we can control the evaporatingtemperature to desired value by just selecting parameters τ_(d) andk_(i).

[0090] It is assumed that the length of two-phase section (or superheat)is controlled to be at the desired value by inlet mass flow rate. FromEquation (8), $\begin{matrix}{\quad {{\overset{.}{m}}_{{i\quad n},i} = \frac{q_{i}}{\left( {1 - x_{0}} \right)h_{\lg}}}} & (14)\end{matrix}$

[0091] Inserting Equation (4) into Equation (12), the control lawbecomes $\begin{matrix}{{\overset{.}{m}}_{out} = {\frac{\sum\limits_{i = 1}^{n}q_{i}}{\left( {1 - x_{0}} \right)h_{\lg}} + {\frac{k}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {{kk}_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}}} & (15)\end{matrix}$

[0092] At the right side of Equation (15), the first term is a nonlinearfunction depending on the states of the system and ambient conditions,and the second and third terms represent a traditional feedback PIcontroller. This nonlinear control provides nonlinear compensation(called feedback linearization) to eliminate the nonlinearity termΣq_(i)/(1−x₀)/h_(lg) in the system dynamics and makes the control designa linear problem.

[0093] Compared to a feedback PI control (currently used inmulti-system) or a self-tuning control, the new nonlinear controlapproach of the invention has the following advantages:

[0094] 1) For a wide range operation,the PI control gains in thisnonlinear control do not need to be tuned adaptively. The feedbacklinearization (nonlinear compensation) provides the adaptation for bigchange of operation conditions. However for a feedback PI controller ora self-tuning control, it is necessary to tune the PI gains fordifferent operating conditions to guarantee stability and performance.

[0095] 2) The new nonlinear control can adapt to the turn on/off ofindoor units accurately and quickly. For example, if j-th indoor unit isturned off from running, we can remove q_(j) from the Equation (15) andthe control input can be changed immediately.

[0096] 3) Since the selection of PI control gains in the new nonlinearcontrol is basically the same as designing a PI control for a knownlinear system, the design procedure is much easier and straghtforward.It can save much time to design control for new product.

[0097] 4) With the nonlinear compensation, The PI gains in the newnonlinear controller can be much smaller.

[0098] In actual operation, the control input to control the evaporatingtemperature is compressor speed. Therefore the mass flow rate {dot over(m)}_(out) is related to the compressor speed ω. The compressor massflow rate is dependent on compressor speed, the low pressure P_(e) andhigh pressure P_(c), and can be expressed by

{dot over (m)} _(out) =ωg(P _(e) ,P _(c))  (16)

[0099] where g(P_(e),P_(c)) can be identified for a given compressor.P_(e) and P_(c) can be measured by two pressure sensors. For a certaintype of the machine tested, $\begin{matrix}{{g\left( {P_{e},P_{c}} \right)} = {\frac{1}{1000}{\left( {{- 0.362} + {0.595P_{e}} + {0.345P_{c}} + {0.207P_{e}^{2}} - {0.073P_{c}^{2}} - {0.019P_{e}P_{c}}} \right).}}} & (17)\end{matrix}$

[0100] In the above equations, the unit for mass flow rate is kg/s, theunit for compressor speed is Hz, and the unit for pressure is MPa. Basedon Equations (15) and (16), the new nonlinear controller to controlevaporating temperature by compressor speed is expressed by$\begin{matrix}{{\omega (t)} = {{\frac{1}{g\left( {P_{e},P_{c}} \right)}\frac{\sum\limits_{i = 1}^{n}{q_{i}(t)}}{\left( {1 - x_{o}} \right)h_{\lg}}} + {\frac{k}{g\left( {P_{e},P_{c}} \right)}\left( {{\frac{1}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {k_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}} \right)}}} & (18)\end{matrix}$

[0101]FIG. 3 contains a schematic diagram of the new nonlinear controlapproach in accordance with the invention. FIG. 4 contains a blockdiagram of the new nonlinear control approach of the invention. In FIG.4, the feedback linearization term is estimated on-line based onmeasured sensor data.

[0102] 1) h_(lg) is calculated from refrigerant thermal property andmeasured evaporating temperature or evaporating pressure.

[0103] 2) Inlet vapor quality x₀ can be estimated as follows duringnominal operation: $\begin{matrix}{x_{0} \approx \frac{{h_{l}\left( P_{c} \right)} - {h_{l}\left( P_{e} \right)}}{h_{\lg}\left( P_{e} \right)}} & (19)\end{matrix}$

[0104] 3) q_(i) can be estimated as the cooling capacity of indoor unitas follows:

q _(i) =K _(h)(ω_(ƒ))*(T _(h) −T _(a))*ƒ(SH)  (20)

[0105] where K_(h)(ω_(ƒ)) is heat transfer coefficient at air side withindoor unit fan speed ω_(ƒ), T_(a) is room air temperature, T_(h) istemperature of heat exchanger, f(SH) is the changing rate of coolingcapacity vs superheat at the outlet of heat exchanger.

[0106] A similar method can be developed to estimate q_(i) based on thedifference between room air temperature and the evaporating temperature.

[0107] It should be pointed out that the estimation for the feedbacklinearization may have some error. The PI control part in the nonlinearcontrol is used to compensate this estimation error.

[0108] 2.2 Simulations for Nonlinear Control of Evaporating Temperature

[0109] Simulation of a multi-system with two indoor units is implementedto demonstrate the performance of the new nonlinear control. Thesimulations described below compare the new control method of theinvention with a feedback PI control shown in FIG. 5, which contains ablock diagram of a linear feedback PI control system.

[0110] Nonlinear Controller:${\overset{.}{m}}_{out} = {\frac{\sum\limits_{i = 1}^{n}q_{i}}{\left( {1 - x_{0}} \right)h_{\lg}} + {\frac{k}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {{kk}_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}}$

[0111] Linear Feedback PI Controller:${\overset{.}{m}}_{out} = {{\frac{k}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {{kk}_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}}$

[0112] For the following simulations, it is assumed that

[0113] Total heat transfer rate of two evaporators${\sum\limits_{i = l}^{2}\quad q_{i}} = {\frac{3.7}{1.7}\left( {T_{a} - T_{e}} \right)}$

[0114] Inlet vapor quality x₀=0.2

[0115] Set point of evaporating temperature T_(e,d)=10° C.

[0116] Initial evaporating temperature T_(e)(t=0)=14° C.

[0117] Room air temperature T_(a)=27° C. $\begin{matrix}{k = {{\sum\limits_{i = 1}^{n}\quad {V_{i}\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}}} = {0.001\left( {{kg}\text{/}{{{^\circ}C}.}} \right)}}} & \quad\end{matrix}$

[0118] h_(lg)≈200 KJ/kg

[0119]FIG. 6A is a graph of time response of evaporating temperaturewith same PI gains for the feedback linearization and PI control of theinvention compared to feedback PI control. FIG. 6A illustrates that withthe same PI gains, the nonlinear control (feedback linearization and PIcontrol) can reach the desired evaporating temperature much fastercompared to the linear feedback PI control. FIG. 6B is a graphillustrating time response of mass flow rate with same PI gains for thefeedback linearization and PI control of the invention compared tofeedback PI control. FIG. 6C is a graph illustrating time response ofcompressor speed with same PI gains for the feedback linearization andPI control of the invention compared to feedback PI control.

[0120]FIG. 7A is a graph of time response of evaporating temperaturewith similar performance for the feedback linearization and PI controlof the invention compared to feedback PI control. FIG. 7A shows thatwith the similar performance, the nonlinear control (feedbacklinearization and PI control) has much smaller PI gains(1/τ_(d)=1.0,k_(i)=0.25) compared to the gains of linear feedback PIcontrol (1/τ_(d)=20,k_(i)=5). Time responses of mass flow rate for thesame case as FIG. 7A is illustrated in FIG. 7B, which is a graph of timeresponse of mass flow rate with similar performance for the feedbacklinearization and PI control of the invention compared to feedback PIcontrol. FIG. 7C is a graph of time response of compressor speed withsimilar performance for the feedback linearization an PI control of theinvention compared to feedback PI control.

[0121] For the next simulation, it is assumed that the estimation offeedback linearization term has 20% error. FIG. 8A is a graph of timeresponse of evaporating temperature where feedback linearization isassumed to have 20% error for same PI gains. FIG. 8A shows that withsame PI gains, the nonlinear control can also reach the desiredevaporating temperature much faster compared to the linear feedback PIcontrol. FIG. 8B is a graph of time response of compressor speed wherefeedback linearization is assumed to have 20% error for same PI gains.

[0122] For the next simulation, it is assumed that the estimation offeedback linearization term has 20% error. FIG. 9A is a graph of timeresponse of evaporating temperature where feedback linearization isassumed to have 20% error for similar performance. FIG. 9B is a graph oftime response of compressor speed where feedback linearization isassumed to have 20% error for similar performance. FIG. 9A illustratesthat with the similar performance, the nonlinear control has muchsmaller PI gains (1/τ_(d)=10,k_(i)=0.25) compared to the gains of linearfeedback PI control (1/τ_(d)=10,k_(i)=2.5).

[0123] For the next simulation, it is assumed that the estimation offeedback linearization term has 20% estimation error for same gains. Thesecond indoor unit is turned off at t=40s and turned on on t=80s. FIG.10A is a graph of time response of evaporating temperature wherefeedback linearization is assumed to have 20% error for same gains andone indoor unit is turned off at t=40s and turned on at t=80s. FIG. 10Bis a graph of time response of compressor speed where feedbacklinearization is assumed to have 20% error for same gains and one indoorunit is turned off at t=40s and turned on at t=80s. FIG. 10A illustratesthat with same PI gains the nonlinear control has much betterperformance compared to linear feedback PI control. It should be notedthat the nonlinear control of the invention can adapt to indoor unitturn on/off with perfect performance, if there is no estimation errorfor feedback linearization.

[0124] For the next simulation, it is assumed that the estimation offeedback linearization term has no estimation error. The second indoorunit is turned off at t=40s and turned on on t=80s. FIG. 11A is a graphof time response of evaporating temperature where feedback linearizationis assumed to have no estimation error and the second indoor unit isturned off at t=40s and turned on at t=80s. FIG. 11B is a graph of thecontrol output of compressor speed where feedback linearization isassumed to have no estimation error and the second indoor unit is turnedoff at t=40s and turned on at t=80s. FIG. 11A illustrates the comparisonof the nonlinear control (1/τ_(d)=1.0,k_(i)=0.25) to feedback PI control(1/τ_(d)=10,k_(i)=2.5). The nonlinear control can adapt to indoor unitturn on/off with perfect performance (Te remains at 10 C).

[0125] 2.3 Nonlinear Control of Superheat

[0126] Equation (4) for dynamics of the length of two-phase section canbe rewritten as $\begin{matrix}{{k\frac{{l(t)}}{t}} = {{- \frac{q}{\left( {1 - x_{0}} \right)h_{\lg}}} + {\overset{.}{m}}_{i\quad n}}} & (21)\end{matrix}$

[0127] where $\begin{matrix}{k = \frac{\rho_{1}\left( {1 - \overset{\_}{\gamma}} \right)}{\left( {1 - x_{0}} \right)}} & (22)\end{matrix}$

[0128] To control the length of two-phase section to a desired valuel_(d) by control input {dot over (m)}_(in), we design the followingnonlinear controller $\begin{matrix}{{\overset{.}{m}}_{i\quad n} = {\frac{q}{\left( {1 - x_{0}} \right)h_{\lg}} + {\frac{k}{\tau_{d}}\left( {{l(t)} - l_{d}} \right)} + {{kk}_{i}{\int{\left( {{l(t)} - l_{d}} \right){t}}}}}} & (23)\end{matrix}$

[0129] By inserting Equation (23) into Equation (21), the controlledl(t) dynamics can be described by the following equation,$\begin{matrix}{\frac{{l(t)}}{t} = {{{- \frac{1}{\tau_{d}}}\left( {{l(t)} - l_{d}} \right)} - {k_{i}{\int{\left( {{l(t)} - l_{d}} \right){t}}}}}} & (24)\end{matrix}$

[0130] where τ_(d) and k_(i) are design parameters representing how fastthe length of two-phase section l(t) reaches the desired set-point l_(d)with guaranteed zero steady state error. From Equation (24) it isobserved that the closed loop dynamics is linear and we can control thelength of two-phase section l(t) to desired value by just selectingparameters τ_(d) and k_(i).

[0131] At the right side of Equation (23), the first term is feedbacklinearization nonlinearly depending on the states of the system andambient conditions which were describe above in section 2.1, and thesecond and third terms represent a traditional feedback PI controller.This nonlinear control provides nonlinear compensation (feedbacklinearization) to eliminate the nonlinearity term q/(1−x₀)/h_(lg) in thesystem dynamics and makes the control design a linear problem. Theadvantages of this new controller are the same as discussed in section2.1.

[0132] In actual operation, the expansion valve openning is used tocontrol superheat value. Therefore it is necessary to relate the massflow rate {dot over (m)}_(in) to the expansion valve openning A_(ν). Theexpansion valve mass flow rate {dot over (m)}_(in) is dependent onexpansion valve openning A_(ν), the low pressure P_(e) and high pressureP_(c), and can be expressed by

{dot over (m)} _(in) =A _(ν) ^(a) g _(ν)(P _(e) ,P _(c))  (25)

[0133] where a and g_(ν)(P_(e),P_(c)) can be identified for a givenexpansion valve. P_(e) and P_(c) can be measured by two pressuresensors. For the indoor unit with the nominal capacity 4 kW tested inDaikin Kanaoko Factory Room 906 in the summer of 2001, a=0.75 and$\begin{matrix}{{g_{v}\left( {P_{e},P_{c}} \right)} = {\frac{1}{10000}\left( {{- 1.38} + {9.07P_{e}} - {0.67P_{c}} - {9.49P_{e}^{2}} + {0.38P_{c}^{2}} + {2.37P_{e}P_{c}}} \right)}} & (26)\end{matrix}$

[0134] In the above equations, the unit for mass flow rate is kg/s, theunit for expansion valve opening is step, and the unit for pressure isMPa.

[0135] Based on Equations (23) and (25), the new nonlinear controller tocontrol the length of two-phase section l(t) by expansion valve isexpressed by $\begin{matrix}{{A_{v}(t)} = \left( {{\frac{1}{g_{v}\left( {P_{e},P_{c}} \right)}\frac{q}{\left( {1 - x_{o}} \right)h_{\lg}}} + {\frac{k}{g_{v}\left( {P_{e},P_{c}} \right)}\left( {{\frac{1}{\tau_{d}}\left( {{l(t)} - l_{d}} \right)} + {k_{i}{\int{\left( {{l(t)} - l_{d}} \right){t}}}}} \right)}} \right)^{\frac{1}{a}}} & (27)\end{matrix}$

[0136] In order to control superheat (SH) value to the desired valueSH_(d) by expansion valve, the following nonlinear control is proposed$\begin{matrix}{{A_{v}(t)} = \left( {{\frac{1}{g_{v}}\frac{q}{\left( {1 - x_{o}} \right)h_{\lg}}} + {\frac{k}{g_{v}}\left( {{\frac{1}{\tau_{d}^{\prime}}\left( {{{SH}(t)} - {SH}_{d}} \right)} + {k_{i}^{\prime}{\int{\left( {{{SH}(t)} - {SH}_{d}} \right){t}}}}} \right)}} \right)^{\frac{1}{a}}} & (28)\end{matrix}$

[0137] where superheat SH(t) can be measured. The SH value can beapproximated by following equation $\begin{matrix}{{SH} \approx {\left( {T_{a} - T_{e}} \right)\left( {1 - ^{\frac{- {c{({L - {l{(t)}}})}}}{\overset{.}{m}}}} \right)}} & (29)\end{matrix}$

[0138] where L is the length of evaporator, and c is a parameter, and{dot over (m)}≈q/(1−x₀)/h_(lg). Since the length of two-phase sectionl(t) can not be directly measured, l(t) can be estimated from SH basedon Equation (29) as follows: $\begin{matrix}{{\hat{l}(t)} = {L + {\frac{\overset{.}{m}}{c}{\ln \left( {1 - \frac{{SH}(t)}{T_{a} - T_{e}}} \right)}}}} & (30)\end{matrix}$

[0139] Therefore the new nonlinear controller expressed by Equation (27)can be expressed by $\begin{matrix}{{A_{v}(t)} = \left( {{\frac{1}{g_{v}\left( {P_{e},P_{c}} \right)}\frac{q}{\left( {1 - x_{o}} \right)h_{\lg}}} + {\frac{k}{g_{v}\left( {P_{e},P_{c}} \right)}\left( {{\frac{1}{\tau_{d}}\left( {{\hat{l}(t)} - l_{d}} \right)} + {k_{i}{\int{\left( {{\hat{l}(t)} - l_{d}} \right){t}}}}} \right)}} \right)^{\frac{1}{a}}} & (31)\end{matrix}$

[0140]FIG. 12 contains a schematic block diagram of nonlinear control ofexpansion valve for l(t) expressed by Equation (31) in accordance withthe invention.

[0141]FIG. 13 contains a schematic block diagram of nonlinear control ofexpansion valve for SH expressed by Equation (28) in accordance with theinvention.

[0142] 2.4 Simulation for Nonlinear Control of Superheat

[0143] Simulation is implemented to demonstrate the performance of thenew nonlinear control of expansion valve in accordance with theinvention. The new control method of the invention is compared with afeedback PI control.

[0144] Controlling the Length of 2-phase Section to the Desired Value:

[0145] Nonlinear Controller:${\overset{.}{m}}_{i\quad n} = {\frac{q}{\left( {1 - x_{o}} \right)h_{\lg}} + {\frac{k}{\tau_{d}}\left( {{l(t)} - l_{d}} \right)} + {{kk}_{i}{\int{\left( {{l(t)} - l_{d}} \right){t}}}}}$

[0146] Linear Feedback PI Controller:${\overset{.}{m}}_{i\quad n} = {{\frac{k}{\tau_{d}}\left( {{l(t)} - l_{d}} \right)} + {{kk}_{i}{\int{\left( {{l(t)} - l_{d}} \right){t}}}}}$

[0147] Controlling Superheat (SH) to the Desired Value:

[0148] Nonlinear Controller:${\overset{.}{m}}_{i\quad n} = {\frac{q}{\left( {1 - x_{o}} \right)h_{\lg}} + {\frac{k}{\tau_{d}}\left( {{{SH}(t)} - {SH}_{d}} \right)} + {{kk}_{i}{\int{\left( {{{SH}(t)} - {SH}_{d}} \right){t}}}}}$

[0149] Linear Feedback PI Controller:${\overset{.}{m}}_{i\quad n} = {{\frac{k}{\tau_{d}}\left( {{{SH}(t)} - {SH}_{d}} \right)} + {{kk}_{i}{\int{\left( {{{SH}(t)} - {SH}_{d}} \right){t}}}}}$

[0150] For the following simulations, one indoor unit is considered, andit is assumed that:

[0151] Heat transfer rate of the evaporator${q(t)} = {\frac{2}{1.7}\left( {T_{a} - T_{e}} \right)\frac{l(t)}{l_{d}}}$

[0152] Inlet vapor quality x₀=0.2

[0153] Evaporating temperature is controlled by compressor to remain atT_(e)=10° C.

[0154] Desired the length of 2-phase sectionl_(d)=0.9L=0.9*8.14(m)=7.32(m)

[0155] Room air temperature T_(a)=27° C.${k = {\frac{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}{\left( {1 - x_{0}} \right)} = 0.0027}},{{{with}\quad \overset{\_}{\gamma}} = 0.93}$

[0156] h_(lg)≈200 KJ/kg

[0157] Desired superheat SH_(d)≈5° C.

[0158] The parameter c in Equation (29) has a value of 0.0053

[0159] For the first simulation case, the length of two-phase section iscontrolled from an initial value 6 m to the desired value 7.32 m. Inthis case, it is assumed that there is no estimation error for feedbacklinearization. FIG. 14A is a graph of the time response of the lengthl(t) of the two-phase section controlled from an initial value of 6 m tothe desired value of 7.32 m, assuming there is no estimation error forfeedback linearization. FIG. 14B is a graph of control input of massflow rate of expansion valve for the case of FIG. 14A. FIG. 14C is agraph of control output of expansion valve opening for the case of FIG.14A. FIG. 14D is a graph of control output of heat flow for the case ofFIG. 14A. FIG. 14E is a graph of control output of superheat for thecase of FIG. 14A. FIG. 14A illustrates that with same PI gains, thenonlinear control can reach the desired length of 2-phase section muchfaster compared to the linear feedback PI control. Time responses ofmass flow rate, expansion valve opening, evaporator heat exchange rate,and superheat value, are shown in FIGS. 14B through 14E.

[0160] For the second simulation case, the length of two-phase sectionis controlled at the desired value 7.32 m. For this case, it is assumedthat indoor fan speed increases such that q becomes${{q(t)} = {{\frac{2b}{1.7}\left( {T_{a} - T_{e}} \right)\frac{l(t)}{l_{d}}\quad b} = {1 + \frac{t}{10}}}},{{t \leq {5s}};{b = 1.5}},{t > {5{s.}}}$

[0161] In this case, it is assumed that there is 20% estimation errorfor feedback linearization. FIG. 15A is a graph of the time response ofthe length l(t) of the two-phase section controlled at the desired valueof 7.32 m, assuming there is 20% estimation error for feedbacklinearization. FIG. 15B is a graph of control input of mass flow rate ofexpansion valve for the case of FIG. 15A. FIG. 15C is a graph of controloutput of expansion valve opening for the case of FIG. 15A. FIG. 15D isa graph of control output of heat flow for the case of FIG. 15A. FIG.15E is a graph of control output of superheat for the case of FIG. 15A.FIG. 15A illustrates that the nonlinear control has much betterperformance compared to the linear feedback PI control. Time responsesof mass flow rate, expansion valve opening, evaporator heat exchangerate, and superheat value, are shown in FIGS. 15B through 15E.

[0162] For the third simulation case, the length of two-phase section iscontrolled from initial value 6 m to the desired value 7.32 m. Assumethere are three different load situations $\begin{matrix}{{q(t)} = {{\frac{K}{1.7}\left( {T_{a} - T_{e}} \right)\frac{l(t)}{l_{d}}\quad (I)K} = 1}} & {{({II})K} = 2} & {{({III})K} = 4}\end{matrix}$

[0163]FIG. 16A is a graph of time response of the length l(t) of thetwo-phase section from and initial value of 6 m to the desired value of7.32 m, assuming three different load situations and no estimation errorfor feedback linearization. FIG. 16B is a graph of control output ofexpansion valve opening for the case of FIG. 16A. FIG. 16A illustratesthat the nonlinear control has the same performance for three loadsituations. However the performance of the linear feedback PI controlhas large difference with the same gains for three different loadsituations. That means PI gains in linear feedback control need to betuned in order to achieve similar good performance for wide rangeoperation. Time response of expansion valve opening is shown in FIG.16B.

[0164] For the fourth simulation case, the superheat is controlled frominitial value 11.3 C to the desired value 5 C. In this case, it isassumed that there is no estimation error for feedback linearization.FIG. 17A is a graph of control output of superheat controlled from aninitial value of 11.3 C to the desired value of 5 C, assuming that thereis no estimation error for feedback linearization. FIG. 17B is a graphof control output of expansion valve opening for the case of FIG. 17A.FIGS. 17A and 17B show the comparison of the nonlinear control and thelinear feedback PI control.

3. Nonlinear Observer To Estimate Heat Transfer Rate

[0165] 3.1. Evaporator Model

[0166] A nonlinear observer is described herein to estimate the heattransfer rate in accordance with the invention. A simplified low-orderevaporator model is used here for the nonlinear observer design. FIG. 18contains a schematic diagram of the low-order evaporator model. T_(e) isthe evaporating temperature. l is the length of the two-phase section.T_(w) is the wall temperature of the tube. T_(a) is the room airtemperature. {dot over (m)}_(in) and {dot over (m)}_(out) are the inletand outlet refrigerant mass flow rates respectively. q is the heattransfer rate from the tube wall to the two-phase refrigerant. q_(a) isthe heat transfer rate from the room to the tube wall.

[0167] Assuming a uniform temperature throughout the evaporator tubewall at the two-phase section, the heat transfer equation of the tubewall is as follow: $\begin{matrix}{{\left( {c_{p}\rho \quad A} \right)_{e}\frac{T_{w}}{t}} = {{\pi \quad D_{0}{\alpha_{0}\left( {T_{a} - T_{w}} \right)}} - {\pi \quad D_{i}{\alpha_{i}\left( {T_{w} - T_{e}} \right)}}}} & (32)\end{matrix}$

[0168] The first term on the right hand side represents the heattransfer rate per unit length from the room to the tube wall. The secondterm represents the heat transfer rate per unit length from the tubewall to the two-phase refrigerant.

[0169] Assuming the mean void fraction {overscore (γ)} is invariant, theliquid mass balance equation in the two-phase section of the evaporatoris $\begin{matrix}{{{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}A\frac{{l(t)}}{t}} = {{- \frac{q}{h_{\lg}}} + {{\overset{.}{m}}_{in}\left( {1 - x_{0}} \right)}}}{and}} & (33) \\{q = {\pi \quad D_{i}\alpha_{i}{l\left( {T_{w} - T_{e}} \right)}}} & (34)\end{matrix}$

[0170] In Eq. (33), the left hand side is the liquid mass change rate inthe evaporator. On the right hand side, q/h_(lg) represents the rate ofliquid evaporating into vapor, and {dot over (m)}_(in) (1−x₀) is theinlet liquid mass flow rate. Eq. (34) can be used to estimate the heattransfer rate.

[0171] The inlet refrigerant mass flow rate {dot over (m)}_(in) isdependent on the expansion valve openning A_(ν), the low pressure P_(e)and high pressure P_(c), and can be expressed by

{dot over (m)} _(in) =A _(ν) ^(a) g _(ν)(P _(e) ,P _(c))  (35)

[0172] where a and g_(ν)(P_(e),P_(c)) can be identified for a givenexpansion valve. P_(e) and P_(c) can be measured by two pressure sensorsor can be estimated from the evaporating temperature and condensingtemperature. For the two-phase section, it is assumed that the pressureis an invariant function of the temperature. Therefore, the inletrefrigerant mass flow rate {dot over (m)}_(in) can be expressed as

{dot over (m)} _(in) =A _(ν) ^(a) g _(ν)(T _(e) ,T _(c))  (36)

[0173] Assuming that the vapor volume is much larger than the liquidvolume in the low-pressure side, the vapor mass balance equation in anevaporator is: $\begin{matrix}{\frac{M_{v}}{t} = {{V\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}\frac{T_{e}}{t}} = {{{\overset{.}{m}}_{in}x_{0}} + \frac{q}{h_{\lg}} - {\overset{.}{m}}_{out}}}} & (37)\end{matrix}$

[0174] where M_(ν)is the total vapor mass and V is the total volume ofthe low-pressure side. h_(g)−h_(l)=h_(lg), where h_(l) and h_(g) arerefrigerant saturated liquid and vapor specific enthalpies. The outletrefrigerant mass flow rate is the same with the compressor mass flowrate which is dependent on compressor speed, the low pressure P_(e) andhigh pressure P_(c), and can be expressed by

{dot over (m)} _(out) =ωg(P _(e) ,P _(c))  (38)

[0175] where g(P_(e),P_(c)) can be identified for a given compressor. Assaid before, the pressure is an invariant function of the temperaturefor the two-phase section. Therefore, the outlet refrigerant mass flowrate can be expressed as

{dot over (m)} _(out) =ωg(T _(e) ,T _(c))  (39)

[0176] Eq. (37) can be written as $\begin{matrix}{\frac{T_{e}}{t} = {{\frac{\pi \quad D_{i}\alpha_{i}}{{kh}_{\lg}}{l\left( {T_{w} - T_{e}} \right)}} + {\frac{x_{0}}{k}{\overset{.}{m}}_{in}} - {\frac{1}{k}{\overset{.}{m}}_{out}}}} & (40)\end{matrix}$

[0177] where$k = {V{\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}.}}$

[0178] 3.2 Equations for Multi-unit Systems

[0179]FIG. 19 contains a schematic diagram of a multi-unit system inaccordance with the present invention.

[0180] Based on Eq. (32), the heat transfer equation for the j-thevaporator can be expressed by $\begin{matrix}{{\left( {c_{p}\rho \quad A} \right)_{e,j}\frac{T_{w,j}}{t}} = {{\pi \quad D_{o,j}{\alpha_{o,j}\left( {T_{a,j} - T_{w,j}} \right)}} - {\pi \quad D_{i,j}{\alpha_{i,j}\left( {T_{w,j} - T_{e}} \right)}}}} & (41)\end{matrix}$

[0181] where j=1,2, . . . ,n, n is the number of indoor units of amulti-unit system.

[0182] Based on Eq. (33), the liquid mass balance for the jth evaporatorcan be expressed by $\begin{matrix}{{{\rho_{l}\left( {1 - {\overset{\_}{\gamma}}_{j}} \right)}A_{j}\frac{{l_{j}(t)}}{t}} = {{- \frac{q_{j}}{h_{\lg}}} + {{\overset{.}{m}}_{{in},j}\left( {1 - x_{0}} \right)}}} & (42)\end{matrix}$

[0183] where q_(j) is the jth evaporator heat transfer rate.

[0184] The vapor mass balance equation for all n evaporators is$\begin{matrix}\begin{matrix}{{\sum\limits_{j = 1}^{n}\frac{M_{v,j}}{t}} = {\sum\limits_{j = 1}^{n}{V_{j}\frac{{\rho_{g}\left( T_{e} \right)}}{T_{e}}\frac{T_{e}}{t}}}} \\{= {{\sum\limits_{j = 1}^{n}{{\overset{.}{m}}_{{in},j}x_{o}}} + \frac{\sum\limits_{j = 1}^{n}q_{j}}{h_{\lg}} - {\overset{.}{m}}_{out}}}\end{matrix} & (43)\end{matrix}$

[0185] where ΣV_(j) is the total volume of the low-pressure side of amulti-unit system.

[0186] 3.2 Nonlinear Observer Design for Evaporator

[0187] Eq. (32), (33), and (40) represent a nonlinear model for theevaporator. These equations can be expressed in the following compactform:

X=ƒ(X,U)  (44)

[0188] where X=[T_(e) T_(w) l]^(T) are state variables, U=[T_(c) ω T_(a)A_(ν)]^(T) are input variables to the evaporator model. This is a highlynonlinear model. Equation (44) can be expressed more explicitly$\begin{matrix}{{\frac{}{t}\begin{pmatrix}T_{e} \\T_{w} \\l\end{pmatrix}} = \begin{pmatrix}{{\frac{\pi \quad D_{i}\alpha_{i}}{{kh}_{\lg}}{l\left( {T_{w} - T_{e}} \right)}} + {\frac{x_{o}}{k}{\overset{.}{m}}_{in}} - {\frac{1}{k}{\overset{.}{m}}_{out}}} \\{{\frac{\pi \quad D_{o}\alpha_{o}}{\left( {c_{p}\rho \quad A} \right)_{e}}\left( {T_{a} - T_{w}} \right)} - {\frac{\pi \quad D_{i}\alpha_{i}}{\left( {c_{p}\rho \quad A} \right)_{e}}\left( {T_{w} - T_{e}} \right)}} \\{{- \frac{\pi \quad D_{i}\alpha_{i}{l\left( {T_{w} - T_{e}} \right)}}{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}\quad A\quad h_{\lg}}} + \frac{{\overset{.}{m}}_{in}\left( {1 - x_{0}} \right)}{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}A}}\end{pmatrix}} & (45)\end{matrix}$

[0189] It is assumed that all input variables can be obtained by directmeasurements. The output of the system is T_(e) which can be obtained bymeasuring the evaporating temperature using a temperature sensor.However, the state variables T_(w) and l can not be measured directly. Anonlinear observer can be designed based on the nonlinear modeldescribed by Equation (45) to estimate T_(w), l and q as follows:$\begin{matrix}{{\frac{}{t}\begin{pmatrix}{\hat{T}}_{e} \\{\hat{T}}_{w} \\\hat{l}\end{pmatrix}} = \begin{pmatrix}{{\frac{\pi \quad D_{i}\alpha_{i}}{{kh}_{\lg}}{l\left( {{\hat{T}}_{w} - {\hat{T}}_{e}} \right)}} + {\frac{x_{o}}{k}{\overset{.}{m}}_{in}} - {\frac{1}{k}{\overset{.}{m}}_{out}} - {L_{1}\left( {{\hat{T}}_{e} - {\hat{T}}_{e}} \right)}} \\{{\frac{\pi \quad D_{o}\alpha_{o}}{\left( {c_{p}\rho \quad A} \right)_{e}}\left( {{\hat{T}}_{a} - {\hat{T}}_{w}} \right)} - {\frac{\pi \quad D_{i}\alpha_{i}}{\left( {c_{p}\rho \quad A} \right)_{e}}\left( {{\hat{T}}_{w} - {\hat{T}}_{e}} \right)} - {L_{2}\left( {{\hat{T}}_{e} - T_{e}} \right)}} \\{{- \frac{\pi \quad D_{i}\alpha_{i}{l\left( {{\hat{T}}_{w} - {\hat{T}}_{e}} \right)}}{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}\quad A\quad h_{\lg}}} + \frac{{\overset{.}{m}}_{in}\left( {1 - x_{0}} \right)}{{\rho_{l}\left( {1 - \overset{\_}{\gamma}} \right)}A} - {L_{3}\left( {{\hat{T}}_{e} - T_{e}} \right)}}\end{pmatrix}} & (46)\end{matrix}$

[0190] where {circumflex over (T)}_(e), {circumflex over (T)}_(w), and{circumflex over (l)} are estimated values based on the nonlinearobserver, Te is the measured evaporating temperature. L1, L2, and L3 areobserver parameters.

[0191] The evaporator heat transfer rate q can not be measured directly.q can be estimated as

{circumflex over (q)}=πD _(i)α_(i) {circumflex over (l)}( {circumflexover (T)} _(w) −{circumflex over (T)} _(e))  (47)

[0192] 3.3 Numerical Simulation and Discussion

[0193] Simulation of a system with one indoor unit is implemented todemonstrate the nonlinear observer design. The initial operating pointis an equilibrium point obtained from the evaporator and compressorequations presented in section 3.1. The initial values are found to be{overscore (l)}=6.0 m, {overscore (T)}_(e)=10.0° C., {overscore(T)}_(w)=12.125° C., {overscore (T)}_(a)=27.0° C., ω=22.363 Hz,{overscore (A)}_(ν)=83.2865step, and {overscore (T)}_(c)=45.0° C.

[0194] The input T_(a) is assumed to be invariant. The other two inputsω and A_(ν)are assumed to change with time such that the inlet mass flowrate and outlet mass flow rate have the profiles as shown in FIGS. 20and 21. FIG. 20 is a graph of inlet mass flow rate input. FIG. 21 is agraph of outlet mass flow rate. The assumed measured output T_(e) isshown in FIG. 22, which is a graph of evaporating temperature assuming ameasurement output value of T_(e). The comparison of Tw and l obtainedfrom the plant model and the nonlinear observer are shown in FIGS. 23and 24. FIG. 23 is a graph of wall temperate Tw showing a comparison ofTw from the model and the nonlinear observer in accordance with theinvention. FIG. 24 is a graph of two-phase length l showing a comparisonof l from the model and the nonlinear observer in accordance with theinvention. It can be seen that the estimated values catch up veryquickly to the values obtained from the model and the estimation errorsare decreased to zero as show in FIGS. 23 and 24.

[0195] By the estimated values of l_(j) and T_(wj) for each evaporator,the evaporator heat transfer rate q_(j) can be obtained by Eq. (47) andtransferred to the global control system.

[0196] In the simulation, the values of observer parameters are L1=0.2,L2=0.5, and L3=0.0078.

4. Applications in Protection Control

[0197] This section describes the applications of the invention inprotection control design of HAVC systems.

[0198] In order to guarantee the operation safety of HVAC systems, it isimportant to control superheat, discharge temperature Td, evaporatingpressure Pe and condensing pressure Pc to be within safety ranges. It isimportant to maintain a certain minimum superheat value for evaporatorto prevent liquid refrigerant from entering the compressor. Thenonlinear control of the invention has much better capability toregulate the superheat value around a given set-point, therefore thesuperheat value can be maintained within a safety range. For example,FIG. 25 illustrates that the nonlinear control has much betterdisturbance rejection capability for protection control of superheat.FIG. 25 is a graph of response of superheat with increase of indoor fanspeed, controlling superheat from 5 C to a desired value 3.5 C for thefeedback linearization and PI control of the invention and feedback PIcontrol. FIG. 25 illustrates the case that indoor fan speed increases.If the indoor fan speed decreases, the feedback PI control will haveundershoot and cannot ensure that the superheat value exceeds a minimumvalue of 3.5 C, however the nonlinear control can guarantee thesuperheat value is maintained around 3.5 C.

[0199] To protect the safety of the compressor, it is important tocontrol the evaporating pressure and condensing pressure to be withinsafe ranges. Nonlinear control of evaporating temperature describedherein ensures that the evaporating pressure can be accuratelycontrolled within the safety range.

[0200] In order to maintain the condensing pressure and dischargetemperature to be within safety ranges, the following protection controlis proposed based on the nonlinear control described herein.

[0201] For purposes of this description, Td,max is the maximum allowabledischarge temperature from the compressor, and Tc,max is the condensingtemperature corresponding to the maximum allowable condensing pressure.

[0202] If Td<0.9Td,max and Tc<0.9Tc,max $\begin{matrix}{{\omega (t)} = {{\frac{1}{g\left( {P_{e},P_{c}} \right)}\frac{\sum\limits_{i = 1}^{n}{q_{i}(t)}}{\left( {1 - x_{o}} \right)h_{\lg}}} +}} \\{{\frac{k}{g\left( {P_{e},P_{c}} \right)}\left( {{\frac{1}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {k_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}} \right)}}\end{matrix}$

[0203] If Td>0.9Td,max or Tc>0.9Tc,max${\omega (t)} = {{\frac{1}{g\left( {P_{e},P_{c}} \right)}\frac{\sum\limits_{i = 1}^{n}\quad {q_{i}(t)}}{\left( {1 - x_{0}} \right)h_{1g}}} + \quad {\frac{k}{g\left( {P_{e},P_{c}} \right)}\left( {{\frac{1}{\tau_{d}}\left( {T_{e} - T_{e,d}} \right)} + {k_{i}{\int{\left( {T_{e} - T_{e,d}} \right){t}}}}} \right)} - {\Delta \quad \omega}}$

 Δω=K _(p1)(T _(d)−0.9T _(d,max))+K _(P2)(T _(c)−0.9T _(c,max))

[0204] where Kp1 and Kp2 are protection control parameters which can beselected based on testing.

[0205] (Kp1=0 if Td<0.9Td,max, Kp2=0 if Tc<0.9Tc,max)

[0206] If Td>Td,max or Tc>Tc,max

ω(t)=ω(t−1 )−K

[0207] Where K is determined by testing.

[0208] It is assumed that the discharge temperature and condensingtemperature are measured by temperature sensors.

[0209] The new feedback linearization approach described herein has mucheasier design procudures and can achieve better control performance forwide range operation including indoor unit turned on/off. Since thenonlinearity in the system dynamics is compensated by the feedbacklinearization, the approach of the invention deals with a PI controllerdesign problem for a known linear system. The simulations describedherein demonstrate that even with large estimation error, the newnonlinear control of the invention can still achieve desiredperformance.

[0210] It is noted that in the foregoing description, the invention isdescribed in terms of a cooling system. It will be understood that theinvention is applicable to a heating configuration also. In that case,the condenser and evaporator are essentially exchanged, in accordancewith known heating configurations. That is, in the heatingconfiguration, the condenser is in thermal communication with the spaceto/from which heat is being transferred, i.e., the condenser is theindoor unit, and the evaporator is the outdoor unit.

[0211] While this invention has been particularly shown and describedwith reference to preferred embodiments thereof, it will be understoodby those skilled in the art that various changes in form and details maybe made therein without departing from the spirit and scope of theinvention as defined by the appended claims.

1. A heat transfer system, comprising: a first heat exchanger; a secondheat exchanger in thermal communication with a space; and a processorfor estimating an amount of heat transferred between the second heatexchanger and the space and altering a control parameter of the heattransfer system based on the estimated amount of heat transferred tocontrol the heat transfer system.
 2. The heat transfer system of claim1, wherein the first heat exchanger is a condenser.
 3. The heat transfersystem of claim 1, wherein the first heat exchanger is an evaporator. 4.The heat transfer system of claim 1, wherein the second heat exchangeris a condenser.
 5. The heat transfer system of claim 1, wherein thesecond heat exchanger is an evaporator.
 6. The heat transfer system ofclaim 5, wherein the processor controls a temperature of refrigerant inthe evaporator.
 7. The heat transfer system of claim 5, wherein theprocessor controls a temperature of refrigerant in the first heatexchanger.
 8. The heat transfer system of claim 5, wherein the processorcontrols a degree of superheat in the evaporator.
 9. The heat transfersystem of claim 8, wherein the parameter altered by the processor is anexpansion valve opening.
 10. The heat transfer system of claim 5,wherein the processor controls a discharge pressure of refrigerant in acompressor of the heat transfer system.
 11. The heat transfer system ofclaim 5, wherein the processor controls a discharge temperature ofrefrigerant in a compressor of the heat transfer system.
 12. The heattransfer system of claim 5, further comprising a plurality ofevaporators in thermal communication with a respective plurality ofspaces.
 13. The heat transfer system of claim 5, further comprising aplurality of evaporators in thermal communication with the space. 14.The heat transfer system of claim 1, further comprising a compressor forincreasing pressure of a refrigerant flowing between the first andsecond heat exchangers.
 15. The heat transfer system of claim 14,wherein the parameter altered by the processor is a speed of thecompressor.
 16. The heat transfer system of claim 1, wherein theprocessor controls a temperature of refrigerant in the second heatexchanger.
 17. The heat transfer system of claim 1, wherein theprocessor controls a temperature of refrigerant in the first heatexchanger.
 18. The heat transfer system of claim 1, wherein theprocessor controls a degree of superheat in the second heat exchanger.19. The heat transfer system of claim 18, wherein the parameter alteredby the processor is an expansion valve opening.
 20. The heat transfersystem of claim 1, wherein the processor controls a discharge pressureof refrigerant in a compressor of the heat transfer system.
 21. The heattransfer system of claim 5, wherein the processor controls a dischargetemperature of refrigerant in a compressor of the heat transfer system.22. The heat transfer system of claim 1, wherein the processor controlsthe parameter using a feedback linearization approach.
 23. The heattransfer system of claim 1, further comprising a plurality of secondheat exchangers in thermal communication with a respective plurality ofspaces.
 24. The heat transfer system of claim 1, further comprising aplurality of second heat exchangers in thermal communication with thespace.
 25. The heat transfer system of claim 1, wherein the heattransfer system is controlled to protect a component of the heattransfer system from damage.
 26. The heat transfer system of claim 25,wherein the component is a compressor.
 27. A method of heat transfer,comprising: providing a first heat exchanger; providing a second heatexchanger in thermal communication with a space; estimating an amount ofheat transferred between the second heat exchanger and the space; andaltering a control parameter based on the estimated amount of heattransferred to control the heat transfer system.
 28. The method of claim27, wherein the first heat exchanger is a condenser.
 29. The method ofclaim 27, wherein the first heat exchanger is a evaporator.
 30. Themethod of claim 27, wherein the second heat exchanger is a condenser.31. The method of claim 27, wherein the second heat exchanger is anevaporator.
 32. The method of claim 31, further comprising controlling atemperature of refrigerant in the evaporator.
 33. The method of claim31, further comprising controlling a temperature of refrigerant in thefirst heat exchanger.
 34. The method of claim 31, further comprisingcontrolling a degree of superheat in the evaporator.
 35. The method ofclaim 34, wherein the parameter altered by the processor is an expansionvalve opening.
 36. The method of claim 31, further comprisingcontrolling a discharge pressure of refrigerant in a compressor of theheat transfer system.
 37. The method of claim 31, further comprisingcontrolling a discharge temperature of refrigerant in a compressor ofthe heat transfer system.
 38. The method of claim 31, further comprisingproviding a plurality of evaporators in thermal communication with arespective plurality of spaces.
 39. The method of claim 31, furthercomprising providing a plurality of evaporators in thermal communicationwith the space.
 40. The method of claim 27, further comprising providinga compressor for increasing pressure of a refrigerant flowing betweenthe first and second heat exchangers.
 41. The method of claim 40,wherein the altered parameter is a speed of the compressor.
 42. Themethod of claim 27, further comprising controlling a temperature ofrefrigerant in the second heat exchanger.
 43. The method of claim 27,further comprising controlling a temperature of refrigerant in the firstheat exchanger.
 44. The method of claim 27, further comprisingcontrolling a degree of superheat in the second heat exchanger.
 45. Themethod of claim 44, wherein the parameter altered by the processor is anexpansion valve opening.
 46. The method of claim 27, further comprisingcontrolling a discharge pressure of refrigerant in a compressor of theheat transfer system.
 47. The method of claim 27, further comprisingcontrolling a discharge temperature of refrigerant in a compressor ofthe heat transfer system.
 48. The method of claim 27, altering theparameter comprises using a feedback linearization approach.
 49. Themethod of claim 27, further comprising providing a plurality of secondheat exchangers in thermal communication with a respective plurality ofspaces.
 50. The method of claim 27, further comprising providing aplurality of second heat exchangers in thermal communication with thespace.
 51. The method of claim 27, wherein the control parameter isaltered to protect a component of the heat transfer system from damage.52. The method of claim 51, wherein the component is a compressor.